$1/f$ Spectrum and 1-Stable Law in One-Dimensional Intermittent Map with Uniform Invariant Measure and Nekhoroshev Stability

We investigate ergodic properties of a one-dimensional intermittent map that has not only an indifferent fixed point but also a singular structure such that a uniform measure is invariant under mapping. The most striking aspect of our model is that stagnant motion around the indifferent fixed point is induced by the log-Weibull law, which is derived from Nekhoroshev stability in the context of nearly-integrable Hamiltonian systems. Using renewal analysis, we derive a logarithmic inverse power decay of the correlation function and a $1/\omega$-like power spectral density. We also derive the so-called 1-stable law as a component of the time-average distribution of a simple observable function. This distributional law enables us to calculate a logarithmic inverse power law of large deviations. Numerical results confirm these analytical results. Finally, we discuss the relationship between the parameters of our model and the degrees of freedom in nearly-integrable Hamiltonian systems.